What is the current best Live Distro for...

1) use on a laptop

2) use on a desktop

3) Should there be 2 answers for each of the above (cd and dvd)?

I have been away from being able to use live distros for a while but in the past it was generally Knoppix that was picked.

There's a hundred "best" answers for both. Depends on what you want to do and how customizable you want to make it.

Which is "best" is a really dumb question. Yes, there are dumb questions. If one was "best" all the others would have fallen into disuse and you wouldn't even have the opportunity to ask it.

The beauty of LiveCDs is that you can download lots of them and try them all until one seems "best" to you.

... for doing what exactly?

well any distro can be installed on laptops and desktops, i would reccomend something like mandriva,suse or ubuntu for you

Debian

They can, but it doesn't mean they won't fail miserably. Laptops are very sensitive to linux, and linux is very sensitive to laptops. Ubuntu 8.04 on a laptop with sis chipset for video ran in 800x600 max, and the wireless network was completely missing, as for the wired one it took an hour too get to broadcast that it actually wants an ip. So at least I know for sure that if your laptop video card has an intel, sis, etc chipset on it, Ubuntu will NOT be the perfect distro for it, as for the wireless network(on a laptop) I'm safe to say that linux in general does not like that.

Fedora Development

apt-get install xserver-xorg-video-sis would have taken care of the sis driver. And Intel works just fine in any distro.

Linux on Laptops
http://www.linux-laptop.net/

Linux Online - Linux on a Laptop
http://www.linux.org/hardware/laptop.html

So why not start with Knoppix and try any other that catches your fancy?
You can get a list of live CD/DVD at Distrowatch.

Personally I love Knoppix, DSL, Astrumi, Super Grub Disk, Parted Magic, Slax, Ubuntu...

Depends on what you need to do.

Almost all of the distros have liveCD version nowadays. Not so sure about liveDVDs though.

Download a couple and see what works well for you. See the top 10 list http://distrowatch.com/dwres.php?resource=major

You need to be more specific. Knoppix is okay. There is also Knoppix-STD and PHLAK if you want a live cd that focuses on security.
There are also live cds that focus on other things, like a live cd that focuses on education - Kinerret (its focus is on arabic and hebrew).

What type of laptop? if you get into trouble trying to run an OS post your laptop info and people will be able to suggest a distro that will suit it or a way to make it work in your laptop.

You can make your own live cd...

Gentoo

I have had a number of live distros that didn't support my laptop when run live but when installed did.

Not sure what thats about but be aware that a Live CD may not provide the full picture on a distro.

Pete